Traces for fractional Sobolev spaces with variable exponents

Abstract

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if p: Ω × Ω → (1,∞) and q : ∂Ω → (1,∞) are continuous functions such that (n − 1)p(x, x) n − sp(x, x) > q(x) in ∂Ω ∩ {x ∈ Ω: n − sp(x, x) > 0}, then the inequality kfkL q(·) (∂Ω) ≤ C n kfkL p¯(·) (Ω) + [f]s,p(·,·) o holds. Here ¯p(x) = p(x, x) and [f]s,p(·,·) denotes the fractional seminorm with variable exponent, that is given by [f]s,p(·,·) := inf λ > 0: Z Ω Z Ω |f(x) − f(y)| p(x,y) λp(x,y) |x − y| n+sp(x,y) dxdy < 1 and kfkL q(·) (∂Ω) and kfkL p¯(·) (Ω) are the usual Lebesgue norms with variable exponent.